Weak stationarity requires constant mean, constant variance, and autocovariance depending only on lag.
Why it matters. Most time-series methods assume it. Prices are non-stationary (they wander); returns are much closer to stationary, which is why analysis is done on returns rather than levels.
Spurious regression is the consequence of ignoring this: two independent random walks regress on each other with high R-squared and significant t-statistics purely because both trend. That result is entirely an artefact, and it is a favourite interview demonstration.